The problem of spherically symmetric electromagnetic radiation E Comay chool of Physics and Astronomy, Raymond and Beverly Sackler Faculry of Exact Sciences. l Aviv University Te/ aviv 69978. Israel Received 18 October 2001; accepted 18 March 2002) Using Maxwell equations for the radiation's transverse fields, it is proved that a genuine spherically symmetric electromagnetic radiation cannot exist. C 2002 American Association of Physics Teachers [DO1:10.119/1.1477432] The existence of a spherically symmetric wave might be ample of constraints imposed by Maxwell equations, and in- simple mathematical structure. Such a structure is obviously can be realized every arrangement of electromagnetic fields regarded as a self-evident phenomenon having a rathe dicates that not true for the longitudinal sound wave emitted from a breath- Let us consider a spherical shell S belonging to the set ng sphere that is placed in an isotropic medium. Also, the mentioned above and examine it at an instant when S in Eq blackbody radiation emitted from an isothermic sphere is (1)does not vanish. Because the electromagnetic fields are statistically spherical. These kinds of waves are not dis- perpendicular to S, they have no radial component More- cussed here. We will discuss waves in the wave zone whose over, because at all points on S, the magnitude of the Poyn distance from the source is much greater than both the ting vector is the same, the magnitude of the transverse field source's linear size and the radiation's wavelength. The prob- E (and B )is the same also. The latter conclusion relies on the lem analyzed here is called"genuine spherically symmetric orthogonality of E and B at the wave zone electromagnetic radiation. This term means that at the wave Consider a point P on the spherical shell S. E(P) denotes Q, e, there exists a set of concentric spherical shells, where the electric field at P. The direction of E(P) determines a great circle on S that passes through P such that the electric field E(P)is tangent to it. Henceforth, this great circle is EXB called the equator(see Fig. 1) Now let us construct a trajectory C on S that is based on the following assumptions. C starts at P. Also for every point is radial and has the same magnitude. (This magnitude var- p on C, the tangent to C at p is in the direction of E(p) ies as the radius of the shell changes and may vary as a Because E has the same value at all points of S, e does not function of time.) The case analyzed in this paper is not restricted to states function defined at every point belonging to S. This defini whose fields have a well-defined parity. Indeed, the Poynting tion of C is unique. Indeed, the tangent unit vector of differ- vector has a well-defined parity S(r)=-S(-r). On the ential geometry takes the following form other hand, the transverse electromagnetic fields E(r) and B(r) are not necessarily related to E(-r)and b(-r),re spectively Unlike longitudinal sound waves, a system emitting a enuine spherically symmetric electromagnetic radiation is where I denotes the arclength. Hence. we obtain a well defined ordinary first-order differential equation. Thus, by far from being trivial. Indeed, an elementary radiating object the existence and uniqueness theorems of differential is either a quantum mechanical system emitting appropriate dipole or higher multipole radiation or a loop of a time- equations, C is well-defined and unique dependent classical current. The latter can also be expanded The trajectory C is used below and the following cases are in multipoles. As is well known, each of these objects emits nalyzed radiation whose intensity varies with direction. However, it(1)One can find a quantity e>0 such that the trajectory C is not clear whether or not one can arrange infinitely many passes at the point o whose distance from the equator is multipoles(each of which emits an infinitesimal amount of greater than E(see Fig. 1) energy) so that the entire result takes the form of a genuine (2)Otherwise pherically symmetric electromagnetic radiation As is well known, electrodynamics examines two kinds of Assume that case(1)holds. Thus after reaching point 2, entities, charges and their currents on one hand and electro- the trajectory C is closed by adding to it the shorter part Cr nagnetic fields on the other. An examination of the possible of the great circle that passes through P and @(the arc ORP ways of building a radiating system containing an arbitrary in Fig. 1). Evidently, the length of C' is shorter than that of number of dipoles (or higher multipoles) looks very difficult. C. The above assumptions imply that the value of the follow- In the following, the fields of a genuine spherically symmet- ric radiation are analyzed, and it is shown that this kind of ing path integral is positive, radiation is inconsistent with Maxwell equations. This con clusion enhances the insight into the internal structure of EdI=EdI+EdI>O electrodynamics. The proof assumes that a genuine spheri- cally symmetric electromagnetic radiation exists and arrives Indeed, the length of C is greater than that of Cand,at at a contradiction. The case discussed here provides an ex- every point of C, the electric field E is tangent to the path Am J Phys. 70(7), July 2002 http://ojps.aiporg/ajp/ O 2002 American Association of Physics Teachers
The problem of spherically symmetric electromagnetic radiation E. Comaya) School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel ~Received 18 October 2001; accepted 18 March 2002! Using Maxwell equations for the radiation’s transverse fields, it is proved that a genuine spherically symmetric electromagnetic radiation cannot exist. © 2002 American Association of Physics Teachers. @DOI: 10.1119/1.1477432# The existence of a spherically symmetric wave might be regarded as a self-evident phenomenon having a rather simple mathematical structure. Such a structure is obviously true for the longitudinal sound wave emitted from a breathing sphere that is placed in an isotropic medium. Also, the blackbody radiation emitted from an isothermic sphere is statistically spherical. These kinds of waves are not discussed here. We will discuss waves in the wave zone whose distance from the source is much greater than both the source’s linear size and the radiation’s wavelength. The problem analyzed here is called ‘‘genuine spherically symmetric electromagnetic radiation.’’ This term means that at the wave zone, there exists a set of concentric spherical shells, where at all points of each shell, the Poynting vector1,2 S5 c 4pEÃB ~1! is radial and has the same magnitude. ~This magnitude varies as the radius of the shell changes and may vary as a function of time.! The case analyzed in this paper is not restricted to states whose fields have a well-defined parity. Indeed, the Poynting vector has a well-defined parity S(r)52S(2r). On the other hand, the transverse electromagnetic fields E(r) and B(r) are not necessarily related to E(2r) and B(2r), respectively. Unlike longitudinal sound waves, a system emitting a genuine spherically symmetric electromagnetic radiation is far from being trivial. Indeed, an elementary radiating object is either a quantum mechanical system emitting appropriate dipole or higher multipole radiation or a loop of a timedependent classical current. The latter can also be expanded in multipoles.3 As is well known, each of these objects emits radiation whose intensity varies with direction. However, it is not clear whether or not one can arrange infinitely many multipoles ~each of which emits an infinitesimal amount of energy! so that the entire result takes the form of a genuine spherically symmetric electromagnetic radiation. As is well known, electrodynamics examines two kinds of entities, charges and their currents on one hand and electromagnetic fields on the other. An examination of the possible ways of building a radiating system containing an arbitrary number of dipoles ~or higher multipoles! looks very difficult. In the following, the fields of a genuine spherically symmetric radiation are analyzed, and it is shown that this kind of radiation is inconsistent with Maxwell equations. This conclusion enhances the insight into the internal structure of electrodynamics. The proof assumes that a genuine spherically symmetric electromagnetic radiation exists and arrives at a contradiction. The case discussed here provides an example of constraints imposed by Maxwell equations, and indicates that not every arrangement of electromagnetic fields can be realized. Let us consider a spherical shell S belonging to the set mentioned above and examine it at an instant when S in Eq. ~1! does not vanish. Because the electromagnetic fields are perpendicular to S, they have no radial component. Moreover, because at all points on S, the magnitude of the Poynting vector is the same, the magnitude of the transverse field E ~and B! is the same also. The latter conclusion relies on the orthogonality of E and B at the wave zone.4 Consider a point P on the spherical shell S. E(P) denotes the electric field at P. The direction of E(P) determines a great circle on S that passes through P such that the electric field E(P) is tangent to it. Henceforth, this great circle is called the equator ~see Fig. 1!. Now let us construct a trajectory C on S that is based on the following assumptions. C starts at P. Also for every point p on C, the tangent to C at p is in the direction of E(p). Because uEu has the same value at all points of S, E does not vanish there and the definition of the tangent is a unique function defined at every point belonging to S. This definition of C is unique. Indeed, the tangent unit vector of differential geometry takes the following form: t[ dr dl 5 E E , ~2! where l denotes the arclength.5 Hence, we obtain a welldefined ordinary first-order differential equation. Thus, by the existence and uniqueness theorems of differential equations,6 C is well-defined and unique. The trajectory C is used below and the following cases are analyzed: ~1! One can find a quantity e.0 such that the trajectory C passes at the point Q whose distance from the equator is greater than e ~see Fig. 1!. ~2! Otherwise. Assume that case ~1! holds. Thus after reaching point Q, the trajectory C is closed by adding to it the shorter part C8 of the great circle that passes through P and Q ~the arc QRP in Fig. 1!. Evidently, the length of C8 is shorter than that of C. The above assumptions imply that the value of the following path integral is positive, R C1C8 E"dl5 E C E"dl1 E C8 E"dl.0. ~3! Indeed, the length of C is greater than that of C8 and, at every point of C, the electric field E is tangent to the path. 715 Am. J. Phys. 70 ~7!, July 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers 715
vector satisfies (VXE)r>0. Hence, Maxwell's equation R (×E)=-c可>0 Thus, the assumption that genuine spherically symmetric radiation exists yields a contradiction, because for any spherical radiation, the longitudinal field component Brto- gether with its derivative with respect to the time, vanishes identically. This result completes the proof that a genuine Fig. 1. A sphere at the wave zone. The thin line denotes a great circle called spherically symmetric electromagnetic radiation cannot he equator. The path Pe is tangent to the equator at point P. The path gRP ist. It can be easily seen that the proof holds not only for an outgoing radiation but for an incoming one as well As stated, there is an infinite number of different systems emitting electromagnetic radiation. Each of these systems is Thus, cos 0 of the scalar product takes the maximal value of a particular arrangement of electric and magnetic dipoles or higher multipoles and appropriate classical currents. Thus unity. Because E is uniform on S, the integral on C is there are infinitely many patterns of electromagnetic radia eater than the absolute value of that of Ch tion. However, in spite of the infinite degrees of freedom econd case Here C coincides with available, not every pattern of radiation can be realized the equator which is taken as the closed path. Hence, in this case C'reduces to a point, but Eq. (3)still holds. Using eLectronic mail: eli@tauphy tau acil vector analysis, we find from Eq(3)that L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields(Perga- mon, Oxford, 1975),. 76 Edl=(V×E)ds>0 D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), P. 237. Seee2,pp.744-74 where S is any surface whose boundary is the closed path D E. Rutherford, Vector Methods(Oliver and Boyd, Edinburgh, 1962),9th C+C. Let us choose the corresponding part of the spherical E. L. ince, Ordinary Differential Equation (Dover, New York, 1956) shell S as the surface S. It follows from Eq. (4)that there exists a region on S where the radial part of the following See Ref 1, p 162, Ref 2, p.657 Am. J. Phys., Vol. 70, No. 7, July 2002 E Comay 716
Thus, cos u of the scalar product takes the maximal value of unity. Because uEu is uniform on S, the integral on C is greater than the absolute value of that of C8. Now, let us turn to the second case. Here C coincides with the equator which is taken as the closed path. Hence, in this case C8 reduces to a point, but Eq. ~3! still holds. Using vector analysis, we find from Eq. ~3! that R C1C8 E"dl5 E S8 ~“ÃE!"ds.0, ~4! where S8 is any surface whose boundary is the closed path C1C8. Let us choose the corresponding part of the spherical shell S as the surface S8. It follows from Eq. ~4! that there exists a region on S8 where the radial part of the following vector satisfies (“ÃE)r.0. Hence, Maxwell’s equation yields ~“ÃE!r52 1 c ]Br ]t .0. ~5! Thus, the assumption that genuine spherically symmetric radiation exists yields a contradiction, because for any spherical radiation, the longitudinal field component Br together with its derivative with respect to the time, vanishes identically.7 This result completes the proof that a genuine spherically symmetric electromagnetic radiation cannot exist. It can be easily seen that the proof holds not only for an outgoing radiation but for an incoming one as well. As stated, there is an infinite number of different systems emitting electromagnetic radiation. Each of these systems is a particular arrangement of electric and magnetic dipoles or higher multipoles and appropriate classical currents. Thus, there are infinitely many patterns of electromagnetic radiation. However, in spite of the infinite degrees of freedom available, not every pattern of radiation can be realized. a! Electronic mail: eli@tauphy.tau.ac.il 1 L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields ~Pergamon, Oxford, 1975!, p. 76. 2 J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1975!, p. 237. 3 See Ref. 2, pp. 744–747. 4 See Ref. 1, p. 162, and Ref. 2, p. 657. 5 D. E. Rutherford, Vector Methods ~Oliver and Boyd, Edinburgh, 1962!, 9th ed., pp. 16–19. 6 E. L. Ince, Ordinary Differential Equation ~Dover, New York, 1956!, Chap. III. 7 See Ref. 1, p. 162, Ref. 2, p. 657. Fig. 1. A sphere at the wave zone. The thin line denotes a great circle called the equator. The path PQ is tangent to the equator at point P. The path QRP is a portion of a great circle ~see the text!. 716 Am. J. Phys., Vol. 70, No. 7, July 2002 E. Comay 716
